So, I added this to the already existing list of what my custom post type supports (omitted the whole thing here) and Page functionalities were now visible on the edit screen; which included templates as well. I created a new template without date, author information, yet with keys for some custm fields. Yet, when I press save, the template goes back to default even though I set it to the one named Song output.

however, the _wp_page_template post meta property never gets set, if you hit save, navigate away and navigate back the drop down is back on default, so would have to hack the code that saves that meta_box's settings to save when the post type isn't a page.